This workshop discusses recent developments both in the study of the properties of initial data for Einstein's equations, and in the study of solutions of the Einstein evolution problem. Cosmic censorship, the formation and stability of black holes, the role of mass and quasi-local mass, and the construction of solutions of the Einstein constraint equations are focus problems for the workshop. We highlight recent developments, and examine major areas in which future progress is likely.

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Students, recent Ph.D.'s, women, and members of underrepresented minorities are particularly encouraged to apply. Funding awards are typically made 6 weeks before the workshop begins. Requests received after the funding deadline are considered only if additional funds become available.

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My talk will be concerned with semilinear wave equations posed on curved manifolds. I will discuss several simple examples illustrating how a non-flat geometry of the domain affects the global behavior of solutions.

Solutions to Einstein's equations with a high degree of symmetry play an important role in guiding our understanding of general relativity. It is therefore natural to ask: to what extent are these solutions representative? In the talk we give examples both of stability and instability and relate the results to more general conjectures. In particular, we illustrate the strong dependence of the conclusions on the presence/absence of a positive cosmological constant

We will present recent results regarding conservation laws for the wave equation on null hypersurfaces. We will show that an important example of a null hypersurface admitting such conserved quantities is the event horizon of extremal black holes. We will also show that a global analysis of the wave equation on such backgrounds implies that certain derivatives of solutions to the wave equation asymptotically blow up along the event horizon of such backgrounds. In the second part of the talk we will present a complete characterization of null hypersurfaces admitting conservation laws. For this, we will introduce and study the gluing problem for characteristic initial data and show that the only obstruction to gluing is in fact the existence of such conservation laws.

The Riemannian positive mass theorem and its rigidity is recently proven for asymptotically flat hypersurfaces in Euclidean space in all dimensions. The rigidity says that the hyperplane is the only asymptotically flat hypersurface with nonnegative scalar curvature whose ADM mass is zero. In this talk, we consider a class of asymptotically flat graphs with small positive mass and show that the graph is effectively close to a hyperplane if the mass is small. This is a joint work with Dan Lee

In this talk I will review the history and use of an information-theoretical inequality introduced by Otto and I at the end of the nineties, which we called the HWI inequality; how it was used recently in some large-dimension results.

In order to control locally a space-time which satisfies the Einstein equations, what are the minimal assumptions one should make on its curvature tensor? The bounded L2 curvature conjecture roughly asserts that one should only need L2 bound on the curvature tensor on a given space-like hypersuface. This conjecture has its roots in the remarkable developments of the last twenty years centered around the issue of optimal well-posedness for nonlinear wave equations. In this context, a corresponding conjecture for nonlinear wave equations cannot hold, unless the nonlinearity has a very special nonlinear structure. I will present the proof of this conjecture, which sheds light on the specific null structure of the Einstein equations. This is joint work with S. Klainerman and I. Rodnianski

This talk will describe recent work with A. Carlotto on a method
for approximating general asymptotically flat initial data sets by ones
which are trivial outside cones of arbitrarily small angle. We will
explain why this type of approximation is optimal in a certain sense, and
we will give some applications of the construction

In this talk I will present old and new results on the construction of solutions to the constraint equations with non-constant mean curvature. From previous work with Mattias Dahl and Emmanuel Humbert, it is known that, on closed manifolds, solutions to the conformal constraint equations exist provided that a certain limit equation admits no non-trivial solution. We extend this result to manifolds with boundary with apparent horizon boundary condition

The Kapustin-Witten equations have been proposed by Witten as a way to obtain knot invariants using gauge-theory. I will report on some initial steps, in collaboration with Witten, to develop the requisite analysis for this program. This involves showing that the natural boundary conditions for these equations (away from the knot) satisfy a nonstandard type of ellipticity, which leads to some uniqueness and regularity results.

The aim of this talk is to provide an overview of recent work concerning local energy and pointwise decay estimates for solutions to the
linear wave equation and for the Maxwell equation on black hole backgrounds. This includes joint work with Jason Metcalfe, Jacob Sterbenz and Mihai Tohaneanu

In the asymptotically locally hyperbolic setting it is possible to have metrics with scalar curvature at least -6 and negative mass when the genus of the conformal boundary at infinity is positive. Using inverse mean curvature flow, we prove a Penrose inequality for these negative mass metrics. The motivation comes from a previous result of P. Chrusciel and W. Simon, which states that the Penrose inequality we prove implies a static uniqueness theorem for negative mass Kottler metrics

Density theorems for the space of asymptotically flat solutions to the Einstein constraint equations have proven useful in various applications. Key in establishing such density theorems are mechanisms to carefully perform prescribed deformations of the constraints mapping. There are further applications to gluing constructions and quasi-local mass. We present an overview of the results, techniques and applications

We consider spacetimes arising from perturbations of the interior of Kerr black holes. These spacetimes have a null boundary in the future such that the metric extends continuously beyond. However, the Christoffel symbols may fail to be square integrable in a neighborhood of any point on the boundary.